Split-Spectrum-Based Distributed State Estimation for Nonlinear Systems With Incremental Quadratic Constraints

This paper investigates the distributed state estimation problem for a class of continuous-time nonlinear systems over sensor networks. The nonlinearities considered are assumed to satisfy the Incremental Quadratic Constraints (<inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>QC), a general framework encompassing Lipschitz, sector-bounded, and slope-restricted nonlinearities. The primary challenge lies in the fact that the system is jointly observable but not locally observable for individual sensors. To address this, we propose a novel distributed observer architecture based on the &#x201C;Split-Spectrum&#x201D; strategy. By performing a coordinate transformation, the estimation error dynamics at each node are decomposed into a locally observable subsystem and an unobservable subsystem. For the observable part, the observer gains are designed by solving Linear Matrix Inequalities (LMIs) derived to ensure robustness against nonlinearities. For the unobservable part, a high-gain consensus protocol is employed to suppress the divergence caused by system dynamics and nonlinear interactions. We provide rigorous stability analysis to show that the estimation errors of all agents converge exponentially to zero, provided the coupling gain is sufficiently large. The feasibility of the proposed design conditions is theoretically guaranteed. Numerical simulations on a chaotic Lorenz system validate the effectiveness of the proposed method, demonstrating its capability in trajectory reconstruction and its robustness to initial errors.